Rotation of a Point Around the Origin
1. The Unit Circle
Definition: The unit circle is a circle of radius 1 centered at the origin $O(0, 0)$.
Its equation is:
$$x^2 + y^2 = 1$$
Every point on the unit circle can be described by its coordinates $(x, y)$, and every such point corresponds to exactly one rotation angle measured from the starting point $P(1, 0)$.
This circle is the foundation for defining all trigonometric functions.
We always begin at the point $P(1, 0)$ — the rightmost point of the unit circle, where the circle meets the positive $x$-axis.
Interactive visualization — available in the full lesson
2. Definition of Rotation by Angle $\alpha$
We define rotation starting from the point $P(1, 0)$ on the unit circle:
If $\alpha > 0$
Move counterclockwise along the circle a distance of $\alpha$ radians.
↺If $\alpha < 0$
Move clockwise along the circle a distance of $|\alpha|$ radians.
↻If $\alpha = 0$
The point stays at $P(1, 0)$.
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Key Idea: The endpoint of this motion is called point $M$ — the point obtained by rotating $P(1, 0)$ by angle $\alpha$ radians.